Let $X$ be a scheme over $S$, and $G$ be an affine group scheme over $S$ acting on $X$. This [Wikipedia article][1] (or also this related [MO question][2]) defines a quotient stack $[X/G]$ as a category of principal $G$-bundles fibred over Sch/S.

When they discuss the map $[X/G] \to X/G$ (when $X/G$ exists as an algebraic space, or let's say scheme for convenience if one prefers), they say "complete the diagram" i.e. if $D\to T$ is a principal $G$-bundle corresponding to a $T$-point $T\to [X/G]$, then we can complete the diagram
$T \leftarrow D\to X \to X/G$ (which I understand as induing map $T \to X/G$)

It is not clear to me how one can induce a map $T\to X/G$ using the fact that $D\to X$ is $G$-equivariant.

**Q1. How do we get the induced map $T\to X/G$?**

**Q2. If we have $T$-point of $X/G$, pull back of the diagram $T\to X/G \leftarrow X$ gives a principal $G$-bundle $D\to T$ and $D\to X$ which is $G$-equivariant. Then what is the obstruction of this map $X/G \to [X/G]$ being the quasi-inverse of $[X/G]\to X/G$?**


  [1]: https://en.wikipedia.org/wiki/Quotient_stack
  [2]: https://mathoverflow.net/questions/159279/understanding-the-definition-of-the-quotient-stack-x-g
  [3]: http://wwwf.imperial.ac.uk/~tsg/Index_files/deformationringsunitarygroup.pdf